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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>: The formal solution to the following initial and boundary value problem</p>
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\begin{equation*}
\begin{aligned}
u_t = \alpha^2u_{xx},&amp;~&amp; 0&lt;x&lt;L,\quad t&gt;0\\
u(x,0) = f(x),&amp;~&amp; 0\leq x\leq L\\
u(0,t)=0,~~~ u(L,t)=0,&amp;~&amp; t&gt;0.
\end{aligned}
\end{equation*}
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<p class="continuation">is</p>
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\begin{equation*}
u(x,t)=\sum_{n=1}^{\infty}u_n(x,t)=\sum_{n=1}^{\infty}C_n e^{-\left(\frac{n\pi\alpha}{L}\right)^2t}\sin\dfrac{n\pi x}{L},
\end{equation*}
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<p class="continuation">where</p>
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\begin{equation*}
C_n=\frac{2}{L}\int_{0}^L f(x)\sin\frac{n\pi x}{L}\textrm{d}x,\qquad n=1,2,3,\cdots.
\end{equation*}
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